MUST an I and O proposition simultaneously be be true?

Question

In the Square of Opposition, MUST I and O proposition "simultaneously be be true" ? The webpage is silent on it.

I think it "COULD be true" as in the square but not a "MUST be true". Can someone show me both a Venn diagram and predicate languages (i.e. with $\forall$ and $\exists$ )

Answer 1

The correct reading is "could" and not "must". The linked page has: "not both false".

See The Traditional Square of Opposition.

The rules are :

Two propositions are contradictory iff they cannot both be true and they cannot both be false [A-O and E-I].

Two propositions are contraries iff they cannot both be true but can both be false [A-E].

Two propositions are subcontraries iff they cannot [not possible] both be false but can both be true [O-I].

"Can be both true": consider the following example : "Some natural number is Even" [I] and "Some natural number is not Even" [O].

"Cannot both be false" means that we have no interpretation where neither the predicate nor its negation apply.

Assuming bivalence, a number is Even or not; thus it is not possible that "Some natural number is Even" [I] and "Some natural number is not Even" [O] are both False.

Can they be one True and one False ? YES, they can.

Consider again the natural numbers starting from zero: "Some natural number is greater-or-equal to zero" is True while "Some natural number is not greater-or-equal (i.e. is less than) to zero" is False.

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Nitpicking comment: Aristotle was also the inventor of Modal logic: "can" means possibility and "must" mean necessity.

The relation between the two is :

“necessarily P” is equivalent to “not possibly not P” and “possibly P” to “not necessarily not P”.